Abstract
In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.
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Bejenaru, I., Ionescu, A. D., Kenig, C. E., Tataru, D.: Global Schrödinger maps in dimensions d ≥ 2: Small data in the critical Sobolev spaces. Annals of Mathematics, 173, 1443–1506 (2011)
Chang, N., Shatah, J., Uhlenbeck, K.: Schrödinger maps. Comm. Pure Appl. Math., 53, 590–602 (2000)
Ding, Q.: A note on NLS and the Schödinger flow of maps. Phys. Lett. A, 248, 49–54 (1998)
Ding, W.: On the Schrödinger Flows, Proc. ICM Beijing, 2002, 283–292
Ding, W., Wang, Y.: Local Schrödinger flow into Kähler manifolds. Sci. China Ser. A, 44(11), 1446–1464 (2001)
Pang, P., Wang, H., Wang, Y.: Schrödinger flow on Hermitian locally symmetric spaces. Comm. Anal. Geom., 10(4), 653–681 (2002)
Rodnianski, I., Rubinstein, Y. A., Staffilani, G.: On the global well-posedness of the one-dimensional Schrödinger map flow. Analysis & PDE, 2(2), 187–209 (2009)
Sulem, P., Sulem, C., Bardos, C.: On the continuous limit for a system of classical spins. Comm. Math. Phys., 107, 431–454 (1986)
Ishimori, Y.: Multi-vortex solutions of a two dimensional nonlinear wave equation. Progr. Theoret. Phys., 72, 33–37 (1984)
Kenig, C. E., Nahmod, A.: The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps. Nonlinearity, 18, 1987–2009 (2005)
Soyeur, A.: The Cauchy problem for the Ishimori equations. J. Funct. Anal., 105, 233–255 (1992)
Kenig, C. E., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math., 134(3), 489–545 (1998)
Ding, W., Yin, H.: Special periodic solutions of Schrödinger flow. Math. Z., 253, 555–570 (2006)
Chen, W., Jost, J.: Maps with prescribed tension fields. Comm. Anal. Geom., 12, 93–109 (2004)
Chen, Q.: Stability and constant boundary-value problems of harmonic maps with potential. J. Austral. Math. Soc. Ser. A, 68(2), 145–154 (2000)
Chen, Q.: Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations. Calc. Var. Partial Differential Equations, 8(2), 91–107 (1999)
Fardoun, A., Ratto, A.: Harmonic maps with potential. Calc. Var. Partial Differential Equations, 5(2), 183–197 (1997)
Fardoun, A., Ratto, A., Regbaoui, R.: On the heat flow for harmonic maps with potential. Ann. Global Anal. Geom., 18(6), 555–567 (2000)
Shatah, J., Struwe, M.: Geometric Wave Equations, Courant Institute of Mathematical Sciences, New York University, New York, 1998
Sterbenz, J., Tataru, D.: Regularity of wave-maps in dimension 2+1. Commun. Math. Phys., 298, 231–264 (2010)
Tao, T.: Global regularity of wave map, III–VII, Arxiv preprints
Hubert, A., Schafer, R.: Magnetic Domains, Springer, Berlin, 1998
Shatah, J., Struwe, M.: The Cauchy problem for wave maps. Int. Math. Res. Not., No. 11, 555–571 (2002)
Gu, C.: On the Cauchy problem for harmonic maps defined on two dimensional Minkowski space. Comm. Pure Appl. Math., 33, 727–737 (1980)
Derdzinski, A., Maschler, G.: Local classification of conformally-Einstein Kähler metrics in higher dimensions. Proc. London Math. Soc. (3), 87(3), 779–819 (2003)
Derdzinski, A., Maschler, G.: Special Kähler-Ricci potentials on compact Kähler manifolds. J. Reine Angew. Math., 593, 73–116 (2006)
Jelonek, W.: Kähler manifolds with quasi-constant holomorphic curvature. Annals of Global Analysis and Geometry, 36(2), 143–159 (2009)
Pedersen, H., Tønnesen-Friedman, C., Valent, G.: Quasi-Eistein Kähler metrics. Lett. Math. Phys., 50(3), 229–241 (1999)
Tian, G., Zhu, X.: Uniqueness of Kähler-Ricci solitons. Acta Math., 184, 271–305 (2000)
Kühnel, W.: Conformal transformations between Einstein spaces. Conformal Geometry (Bonn, 1985/1986), Aspects of Math., Vol. E12, Vieweg, Braunschweig, 1988, 105–146
Kobayashi, S.: Transformation Groups in Differential Geometry, Springer-Verlag, New York-Heidelberg, 1972
Huang, P., Wang, Y.: Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere. Preprint
Zhou, Y.: Global weak solutions for 1 + 2 dimensional wave maps into homogeneous spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 16(4), 411–422 (1999)
Müller, S., Struwe, M.: Global existence of wave maps in 1 + 2 dimensions with finite energy data. Topol. Methods Nonlinear Anal., 7(2), 245–259 (1996)
McGahagan, H.: An approximation scheme for Schrödinger maps. Comm. Partial Differential Equations, 32(1–3), 375–400 (2007)
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Partially supported by 973 Project of China (Grant No. 2006CB805902)
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Song, C., Wang, Y.D. Schrödinger soliton from Lorentzian manifolds. Acta. Math. Sin.-English Ser. 27, 1455–1476 (2011). https://doi.org/10.1007/s10114-011-0229-y
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DOI: https://doi.org/10.1007/s10114-011-0229-y